Principle of testing LC resonance frequency applied to PLL

Research background
Traditionally, the test method for LC resonant frequency is to find the frequency point with maximum output by changing the frequency of the signal added to the LC resonant circuit (directly or indirectly) point by point, and define this frequency point as the LC resonant frequency. Obviously, the disadvantages of this test method are: the test method is relatively complicated, the test time is long, the test accuracy is low, and it is directly affected by the temperature change of the resonant body, especially the resonant body containing the magnetic core due to the long test time. The principle and method of testing the LC resonant frequency based on PLL to be introduced in this paper are fast, highly accurate, and not affected by temperature changes, and also have the characteristics of simple test methods. This paper mainly simplifies the principle of using PLL to test the LC resonant frequency from a theoretical perspective.

The basic principle
test of LC resonant frequency can be completed through the secondary coupling loop form shown in Figure 1. Among them, L2C2 forms an LC resonant loop to be tested, L1 is the transmitting coil, and Li is a receiving coil with only a single turn. Generally, the test conditions can be met: 1/ωCi》Ri》ω Li, M2》M1》M3. Here ω is the actual working angular frequency, Ri, Ci and R1, C1 are the circuit parameters of the access loop between the receiving coil and the transmitting coil, M2 is the coupling coefficient between the LC resonant loop to be tested and the receiving coil, M1 is the coupling coefficient between the LC resonant loop to be tested and the transmitting coil, and M3 is the coupling coefficient between the transmitting coil and the receiving coil. It can be concluded from Figure 1 that the above test conditions are met.

Here V1 is the voltage of the transmitted signal, V2 is the voltage of the received signal, and the transfer function of the test loop is determined by the following formula.

Figure 1 Schematic diagram of LC resonance test circuit

According to the properties of the LC resonant circuit, we can get:

Here, ω01 and ω02 (ω02>ω01) are the resonant angular frequencies of the transmitting coil LC resonant circuit and the LC resonant circuit to be measured. In practical applications, Q2 is around 100, while Q1 is less than 1. At this time, Equation 3 can be simplified to:

Of which:

If the angular frequency difference Δω between the actual working angular frequency and the resonant angular frequency to be measured is controlled to be much smaller than ω02, the higher-order terms of Δω can be ignored and Equation 5 can be further simplified.

Therefore, the amplitude function and phase function can be simplified as follows respectively.

The amplitude is maximum at resonance, and the amplitude and phase functions can be given by the following equations respectively.

When the circuit parameters are set to f02=83kHz, f01=800Hz, Q1=0.1, Q2=70, M2/L2=0.2, M1/L1=0.1, M1/L2=10-4, and R 2/Ri=10-3, the simulation results under Mathcad are shown in Figure 2, where the amplitude characteristic is the normalized characteristic, d=Δf. It can be seen from the figure that there is a cycle skip phenomenon, which is caused by the zeroing of the denominator part of the fraction in Formula 8. The angular frequency difference at this point is defined as Δω0, and the following relationship can be obtained from Formula 8.

The above formula satisfies the PLL working condition, that is, the PLL loop using the above phase signal is finally locked on the resonant frequency of the LC loop to be tested. In actual use, it is easy to achieve A, B < 1 according to the test conditions. The sensitivity of the loop composed of Figure 1 is:

[2] Therefore, even if ω02 changes significantly, θ0 changes very little, that is, θ0 can be regarded as a constant. Below we only discuss the operation of the PLL in the vicinity of the resonant angular frequency. Then, equation 8 is changed as follows.

The phase is compensated using equation 10. After compensation, the actual phase Δθ of equation 8 is:

The compensation method can generally use a bidirectional delay loop. Since a fixed delay loop is used in actual applications, there will be an error between the measured resonant angular frequency and the actual resonant angular frequency. Assume that the difference in the resonant angular frequency is Δω02, and the operating angular frequency range satisfies Δω < ω 02. Assuming that the actual phase to be compensated is determined by equation 10, a phase difference Δθ0 will be generated with the fixed phase compensation. Then, equation 10 can be used to obtain the following relationship.

Here Δω 02 is the resonant angular frequency difference caused by the fixed delay when the PLL loop works in the locked state. For a specific LC resonant loop, this is a constant. When the test conditions are met, even if a fixed phase is introduced for compensation, the error generated is very small and can be almost ignored.

Example
Based on the principle introduced in this article, an LC resonance frequency tester was trial-made to test 6 types of LC resonance body samples (products). This LC resonance tester has been applied to the production and product inspection of 6 types of LC resonance bodies. For specific results, please refer to Table 1 and Table 2.

.Conclusion
This paper proposes a principle for testing the LC resonant frequency quickly, with high accuracy and without being affected by temperature changes, and verifies the effectiveness of the above principle through specific examples. A future research topic is how to clarify the differences between the PLL operating characteristics when the transmitting and receiving ends are in the same frequency state and the PLL operating characteristics when the transmitting and receiving ends are in different frequency states.

References 
[1] "High Frequency Circuits" by Tsinghua University Communication Research Group, first edition, 1979, People's Posts and Telecommunications Press 
[2] "Analog Integrated Circuit Design Technology" by PR Gray and RG Meyer, translated by Nagata Takashi, 1990, Beifukan